Complex Numbers: Finding Real Part of a Ratio

This problem involves complex numbers, specifically exploring the relationship between their sum, difference, modulus, and the real part of their ratio.

Understanding the Condition

We are given two complex numbers, \(z_1\) and \(z_2\). The main condition provided is:

\(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\)

This equation relates the magnitudes (moduli) of the sum and difference of the two complex numbers. Let's break down what this means.

Step-by-Step Derivation

• Step 1: Use Modulus Properties

  The property \(|a/b| = |a|/|b|\) allows us to rewrite the condition as:

  \(\frac{|z_1+z_2|}{|z_1-z_2|} = 1\)

  Multiplying both sides by \(|z_1-z_2|\) (assuming \(z_1 \neq z_2\), otherwise the denominator is zero), we get:

  \(|z_1+z_2| = |z_1-z_2|\)

• Step 2: Square Both Sides

  To work with the complex numbers algebraically, we square both sides:

  \(|z_1+z_2|^2 = |z_1-z_2|^2\)

• Step 3: Apply \(|w|^2 = w \cdot \bar{w}\)

  We use the property that the square of the modulus of a complex number \(w\) is equal to the product of the number and its conjugate (\(\bar{w}\)). Applying this to both sides:

  \((z_1+z_2)(\overline{z_1+z_2}) = (z_1-z_2)(\overline{z_1-z_2})\)

  Using the property that the conjugate of a sum/difference is the sum/difference of the conjugates (\(\overline{a+b} = \bar{a}+\bar{b}\) and \(\overline{a-b} = \bar{a}-\bar{b}\)):

  \((z_1+z_2)(\bar{z_1}+\bar{z_2}) = (z_1-z_2)(\bar{z_1}-\bar{z_2})\)

• Step 4: Expand and Simplify

  Now, expand both sides of the equation:

  Left Side: \(z_1\bar{z_1} + z_1\bar{z_2} + z_2\bar{z_1} + z_2\bar{z_2}\)

  Right Side: \(z_1\bar{z_1} - z_1\bar{z_2} - z_2\bar{z_1} + z_2\bar{z_2}\)

  Equating the expanded forms:

  \(z_1\bar{z_1} + z_1\bar{z_2} + z_2\bar{z_1} + z_2\bar{z_2} = z_1\bar{z_1} - z_1\bar{z_2} - z_2\bar{z_1} + z_2\bar{z_2}\)

  Cancel out the terms \(z_1\bar{z_1}\) and \(z_2\bar{z_2}\) from both sides:

  \(z_1\bar{z_2} + z_2\bar{z_1} = -z_1\bar{z_2} - z_2\bar{z_1}\)

  Rearrange the terms to one side:

  \(2(z_1\bar{z_2} + z_2\bar{z_1}) = 0\)

  This simplifies to:

  \(z_1\bar{z_2} + z_2\bar{z_1} = 0\)

• Step 5: Relate to the Target Expression

  We need to find the value of \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\). Let's manipulate the equation from Step 4 to involve the ratio \(\frac{z_1}{z_2}\).

  Divide the equation \(z_1\bar{z_2} + z_2\bar{z_1} = 0\) by \(|z_2|^2 = z_2\bar{z_2}\) (assuming \(z_2 \neq 0\)):

  \(\frac{z_1\bar{z_2}}{z_2\bar{z_2}} + \frac{z_2\bar{z_1}}{z_2\bar{z_2}} = 0\)

  Simplify the terms:

  \(\frac{z_1}{z_2} \cdot \frac{\bar{z_2}}{\bar{z_2}} + \frac{z_2}{z_2} \cdot \frac{\bar{z_1}}{\bar{z_2}} = 0\)

  \(\frac{z_1}{z_2} + 1 \cdot \frac{\bar{z_1}}{\bar{z_2}} = 0\)

  Recognize that \(\frac{\bar{z_1}}{\bar{z_2}}\) is the conjugate of \(\frac{z_1}{z_2}\), i.e., \(\frac{\bar{z_1}}{\bar{z_2}} = \overline{\left(\frac{z_1}{z_2}\right)}\).

  So the equation becomes:

  \(\frac{z_1}{z_2} + \overline{\left(\frac{z_1}{z_2}\right)} = 0\)

• Step 6: Use Real Part Definition

  Recall that for any complex number \(w\), \(w + \bar{w} = 2 \cdot \text{Re}(w)\).

  Let \(w = \frac{z_1}{z_2}\). Then the equation from Step 5 is \(w + \bar{w} = 0\).

  Applying the real part definition:

  \(2 \cdot \text{Re}\left(\frac{z_1}{z_2}\right) = 0\)

  Dividing by 2 gives:

  \(\text{Re}\left(\frac{z_1}{z_2}\right) = 0\)

• Step 7: Calculate the Final Expression

  The question asks for the value of \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\).

  Substituting the result from Step 6:

  \(\text{Re}\left(\frac{z_1}{z_2}\right)+1 = 0 + 1 = 1\)

Conclusion

Based on the derivation from the given condition \(\left|\frac{z_1+z_2}{z_1-z_2}\right| = 1\), we found that the real part of the ratio \(\frac{z_1}{z_2}\) is 0. Therefore, \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) equals 1.